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Full text of "Treatise On Analysis Vol-Ii"

226       XIII    INTEGRATION

(13.21.9)    A ^-measurable function h is v-integrable if and only if

This follows from (13.21.8) and (13.9.13).

(13.21.10)    Let A be a ^-measurable set in X x Y.

(i)    The set M of all x e X such that the section A(x) is not \i-measurable is
^-negligible', the function jth-/^*(A(.x)) is ^-measurable; and

*(A) = J V

In particular, ifA(x) is ^-negligible except on a ^-negligible set of values ofx,
then A is ^-negligible.

(ii)   If A is v-integrable, then the set of all x e X such that A(x) is not
\L-integrable is ^-negligible; the function x\-*/j,(A(x)) (which is defined almost

everywhere with respect to X) h X-integrable; and v(A) = \fi(A(x)) dk(x).
These assertions are particular cases of (13.21.6), (13.21.8) and (13.21.7).

(13.21.11).   Letf(resp.g) be a mapping ofX (resp. Y) into [0, +00]. With
the convention of (1 3.11 ) for products, we have

= ( f *

\J

(13.21.11.1)         /(xfoOO d%x) dp(y) =    f */(x)

By virtue of (13.21.4), we have

jj /W000 d%x) dn(y) ^ f * dl(x) Ff(x)g(y) dn(y).

On the other hand, for each x e X we have

(

and

f*#

x) J *f(x)g(y) dn(y) = f *( (

g(y)(13.21.7) the functions x^fax.y) d^(y) are A-integrable and we have